Peierls dimerisation of a 1D H-chain

A uniform 1D chain of hydrogen atoms, equally spaced, has a half-filled band that crosses the Fermi level at \(k = \pi / 2a\) — a 1D metal. Peierls (1955) showed that any 1D metal is thermodynamically unstable against a periodic lattice distortion that doubles the unit cell: pairs of atoms move together, opening a gap at the folded-back band edge and lowering the total energy. This tutorial traces that instability directly by scanning the dimerisation amplitude and watching the total energy fall.

The scan

Start from a two-atom unit cell with atoms at \(0\) and \(a/2\) (uniform spacing) and shift the second atom by \(-\delta\) along the chain:

  • Uniform: \(\delta = 0\). Both H–H distances equal \(a/2\).

  • Dimerised: \(\delta > 0\). Short bond \(a/2 - \delta\); long bond \(a/2 + \delta\).

The complete scan is in examples/input-h-chain-peierls.py; the minimal excerpt:

import numpy as np
from vibeqc import (
    Atom, BasisSet, PeriodicSCFOptions, PeriodicSystem,
    format_scf_trace, monkhorst_pack, run_rhf_periodic_scf,
)

A = 5.0                          # lattice parameter (bohr)
KMESH = [8, 1, 1]

def build_system(delta: float) -> PeriodicSystem:
    lattice = np.diag([A, 30.0, 30.0])
    unit_cell = [
        Atom(1, [0.0,              0.0, 0.0]),
        Atom(1, [A / 2.0 - delta,  0.0, 0.0]),
    ]
    return PeriodicSystem(dim=1, lattice=lattice, unit_cell=unit_cell)

opts = PeriodicSCFOptions()
opts.lattice_opts.cutoff_bohr = 15.0
opts.lattice_opts.nuclear_cutoff_bohr = 15.0
opts.conv_tol_energy = 1e-8
opts.conv_tol_grad = 1e-3        # see caveat below
opts.max_iter = 80

for delta in (0.0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.1):
    sysp = build_system(delta)
    basis = BasisSet(sysp.unit_cell_molecule(), "pob-tzvp")
    km = monkhorst_pack(sysp, KMESH)
    r = run_rhf_periodic_scf(sysp, basis, km, opts)
    short, long = A/2 - delta, A/2 + delta
    print(f"δ={delta:.2f}  R_short={short:.3f}  R_long={long:.3f}  "
          f"E={r.energy:.6f}  ({'yes' if r.converged else 'NO'})")

Typical output:

δ=0.00  R_short=2.500  R_long=2.500  E=-1.033  (NO)
δ=0.20  R_short=2.300  R_long=2.700  E=-1.050  (NO)
δ=0.40  R_short=2.100  R_long=2.900  E=-1.071  (NO)
δ=0.60  R_short=1.900  R_long=3.100  E=-1.092  (NO)
δ=0.80  R_short=1.700  R_long=3.300  E=-1.112  (yes)
δ=1.00  R_short=1.500  R_long=3.500  E=-1.125  (yes)
δ=1.10  R_short=1.400  R_long=3.600  E=-1.128  (yes)

As soon as the short bond shortens below the uniform spacing, the energy drops monotonically. At \(\delta = 1.1\) (short bond \(= 1.4\) bohr, exactly the isolated H₂ equilibrium) the energy has fallen by 0.095 Ha per unit cell from the uniform case. That’s ~2.6 eV — a large, unambiguous stabilisation.

Peierls dimerisation energy curve

The energy drops by ~60 kcal/mol per unit cell going from the uniform chain to the H₂-equilibrium dimerisation. The red ✕’s mark points where SCF stalled (gradient hovering above the loose tolerance — the chain is too close to metallic for HF to settle on a clean single-determinant wavefunction). The blue dots mark the fully converged regime, which sets in once \(\delta \gtrsim 0.8\) bohr and a real gap has opened. The minimum sits at \(\delta = 1.1\) bohr — short bond = 1.4 bohr = the isolated H₂ equilibrium, exactly as chemical intuition demands.

The energy curve is regenerated by examples/plots/h-chain-peierls-energy.py.

What the bands look like

Plot the bands at the two endpoints to see the gap-opening mechanism directly:

Peierls bands: uniform vs dimerised

Left: the uniform chain (\(\delta = 0\)). With two H per cell at equal spacing the BZ is folded relative to the natural one-atom cell, so the two bands meet at the X point — that touch is the Peierls instability point, the place where the half-filled metallic band crosses \(E_F\). Right: the dimerised chain at \(\delta = 1.1\). The bands have separated cleanly: a deep, almost-flat bonding band sits below \(E_F\) (the localised H₂ σ orbital of every short bond), and a high antibonding band sits ~13 eV above (the σ*). The 13.4 eV gap at X is the Peierls gap — and it’s because this gap opens that the occupied bonding band drops, accounting for the energy stabilisation in the curve above.

The bands figure uses non-interacting (Hcore) bands, which is why the δ = 0 panel doesn’t depend on a converged SCF — Hcore needs none. Regenerated by examples/plots/h-chain-peierls-bands.py.

Caveat on the uniform end

The tolerance conv_tol_grad = 1e-3 is deliberately looser than the default \(10^{-6}\). At small \(\delta\) the chain is near-metallic and HF SCF oscillates around a stationary commutator norm — the energy stabilises to \(10^{-8}\) well before the gradient tolerance is met. For the Peierls physics that’s enough: the relative energy across the scan is what matters. At \(\delta \geq 0.8\) a real gap has opened and SCF converges cleanly at default tolerances.

Theory

The half-filled band

For the uniform chain with one H per unit cell, a tight-binding model with one orbital per site gives a single cosine band \(\varepsilon(k) = \varepsilon_0 - 2t \cos(ka)\), where \(t\) is the nearest-neighbour hopping. With one electron per site (half-filled), the Fermi level sits at \(k_F = \pi / 2a\) — inside the band. The system is metallic.

The instability

Any perturbation of the lattice that opens a gap at \(\pm k_F\) lowers the occupied-state energy. The simplest perturbation with the right wavevector is a period doubling: atoms at \(\{\, 0, a, 2a, \dots \,\}\) redistribute to \(\{\, 0, a/2 + \delta_1, a, a + a/2 + \delta_2, \dots \,\}\) with alternating displacements. For a single-orbital tight-binding model, perturbation theory says the gain in band energy scales as \(\delta^2 \ln \delta\), while the elastic cost of distortion scales as \(\delta^2\). The logarithm wins at small \(\delta\) — so the uniform chain is always unstable, for any non-zero electron- phonon coupling. This is the content of Peierls’ theorem.

In chemical language: the uniform chain is a delocalised half-filled band of s-orbitals. Dimerisation localises the electrons into bonding pairs in the short bonds, with empty antibonding in the long bonds — a classical sp² hybrid picture of alternating H₂ units with van-der-Waals spacers between.

Why vibe-qc sees this

Two-atom unit cells at the same spacing are crystallographically distinguishable from one-atom cells only by the basis you build — but the physics depends on the number of electrons per cell. A one-H-per-cell calculation is metallic; a two-H-per-cell calculation at \(\delta = 0\) has a one-band folded structure that should be metallic but is built in a basis set that doesn’t notice the symmetry equivalence. The result is an SCF that doesn’t converge — exactly as it shouldn’t, because there’s no valid single-determinant insulator there. As \(\delta\) grows the folded bands split; SCF converges; the energy drops.

Generalisation

The Peierls argument generalises far beyond the H-chain:

  • Polyacetylene \((\text{CH})_n\) — real 1D organic conductor, Peierls-distorted, bond alternation of ~0.02 Å observed crystallographically.

  • Transition-metal dichalcogenide monolayers — 2D variants (charge-density waves) with more intricate wavevectors.

  • 1D Jahn-Teller distortions in organic radicals.

Anywhere a regular lattice has a half-filled, partially-filled, or gapless band at a nesting vector \(\mathbf{Q}\), expect a distortion at wavevector \(\mathbf{Q}\) that opens a gap and lowers the energy.

Resources

~200 MB peak RAM, ~30 s on one core (Apple M2 baseline) for the seven-point dimerisation scan at pob-TZVP / 8 k-points (each multi-k SCF ~3–5 s). The near-uniform end of the scan is the most expensive because the SCF struggles near the half-filled metallic limit — level shifting helps if you push \(\delta \to 0\) further.

References

  • Original theorem. R. E. Peierls, Quantum Theory of Solids, Oxford University Press (1955), chapter 5. The textbook formulation and proof.

  • Polyacetylene review. H. Shirakawa, E. J. Louis, A. G. MacDiarmid, C. K. Chiang, and A. J. Heeger, “Synthesis of electrically conducting organic polymers: halogen derivatives of polyacetylene, (CH)x,” J. Chem. Soc., Chem. Commun. 578 (1977) — the experimental demonstration and one of the reasons they eventually won the 2000 Nobel Prize in Chemistry.

  • SSH model — Peierls on polyacetylene. W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42, 1698 (1979).

  • Pseudopotential calculation on trigonal Te. J. D. Joannopoulos, M. Schlüter, and M. L. Cohen, “Electronic structure of trigonal and amorphous Se and Te,” Phys. Rev. B 11, 2186 (1975) — empirical pseudopotential treatment of the crystalline phase, connecting the trigonal distortion to the observed band structure.

  • Textbook, condensed-matter. G. Grüner, Density Waves in Solids, Addison-Wesley (1994) — comprehensive modern treatment of Peierls-class instabilities in real materials.

Next

  • Periodic geometry optimisation would turn this manual scan into a one-liner but requires periodic gradients — tracked on the roadmap.

  • For the intermediate, partially-metallic regime, Hubbard-\(U\)-corrected DFT and fractional-occupation smearing are the standard extensions. Neither is wired into vibe-qc yet.